Properties

Label 100325.a
Sato-Tate group $\mathrm{USp}(4)$
\(\End(J_{\overline{\Q}}) \otimes \R\) \(\R\)
\(\End(J_{\overline{\Q}}) \otimes \Q\) \(\Q\)
\(\overline{\Q}\)-simple yes
\(\mathrm{GL}_2\)-type no

Related objects

Learn more about

Genus 2 curves in isogeny class 100325.a

Label Equation
100325.a.100325.1 \(y^2 + x^2y = x^5 - 2x^4 + 4x^3 - 4x^2 + 3x - 1\)

L-function data

Analytic rank:\(1\)
 
Bad L-factors:
Prime L-Factor
\(5\)\( 1 + 2 T + 5 T^{2}\)
\(4013\)\( ( 1 - T )( 1 + 36 T + 4013 T^{2} )\)
 
Good L-factors:
Prime L-Factor
\(2\)\( ( 1 - 2 T + 2 T^{2} )( 1 + T + 2 T^{2} )\)
\(3\)\( ( 1 - 2 T + 3 T^{2} )( 1 + T + 3 T^{2} )\)
\(7\)\( 1 + 3 T + 13 T^{2} + 21 T^{3} + 49 T^{4}\)
\(11\)\( 1 - 3 T + 11 T^{2} - 33 T^{3} + 121 T^{4}\)
\(13\)\( 1 - 3 T + 7 T^{2} - 39 T^{3} + 169 T^{4}\)
\(17\)\( 1 - 5 T + 15 T^{2} - 85 T^{3} + 289 T^{4}\)
\(19\)\( 1 - T + 2 T^{2} - 19 T^{3} + 361 T^{4}\)
\(23\)\( 1 - 4 T + 16 T^{2} - 92 T^{3} + 529 T^{4}\)
\(29\)\( 1 + 8 T + 42 T^{2} + 232 T^{3} + 841 T^{4}\)
$\cdots$$\cdots$
 
See L-function page for more information

Sato-Tate group

\(\mathrm{ST} =\) $\mathrm{USp}(4)$

Endomorphisms of the Jacobian

Not of \(\GL_2\)-type over \(\Q\)

All \(\overline{\Q}\)-endomorphisms of the Jacobian are defined over \(\Q\).

More complete information on endomorphism algebras and rings can be found on the pages of the individual curves in the isogeny class.